A solution is provided to the problem of finding the probability distribution of elements of a random matrix in terms of the distribution of eigenvalues and eigenvectors. It is then proved that completely isotropic eigenvectors can become localized when the eigenvalues increase exponentially. This general result confirms the prediction of a spontaneous breaking of the unitary transformation, U(N), invariance of random matrix ensembles, in the limit of extremely soft confinement. An algorithm is implemented to generate eigenvectors with broken symmetry. The theory is then verified numerically.

• Received 12 August 1999

DOI:https://doi.org/10.1103/PhysRevE.61.R3291